♣

Introduction to Shafarevich Conjecturefor Projective Calabi-Yau Varieties

Second Version

Yi Zhang 1

Center of Mathematical Sciences atZhejiang University

&The Institute of Mathematical Sciences,The Chinese University of Hong Kong

1Email:yzhang@cms.zju.edu.cn; yzhang@ims.cuhk.edu.hk

Abstract

In this notes,we introduce some interesting and important top-ics in algebraic geometry: Shafarevich conjecture over functionfield,moduli space of polarized projective Calabi-Yau manifoldsand the analogue Shafarevich Conjecture of families of Calabi-Yau manifolds. The author taught some parts of the notes in ashort course of the 2003 summer school held in Center of Math-ematical Sciences at Zhejiang University.

Contents

1. Introduction to Shafarevich Conjecture 11.1. Shafarevich conjecture over function field 11.2. The case of higher dimensional fibers 21.3. Appendix: Arithmetic version of Shafarevich

conjecture 62. The Modern Proof of Original Shafarevich Conjecture 82.1. Moduli of Curves 82.2. Rigidity property 92.3. Boundedness 103. Moduli Properties of Families of Calabi-Yau Manifolds 133.1. Family of polarized algebraic manifolds 133.2. Deformation theory and Torelli theorem 163.3. Appendix: Fundamental results of Calabi-Yau

manifolds 194. Rigidity on Families of Calabi-Yau manifolds 224.1. Endomorphism of Higgs bundles over product

varieties 224.2. Rigid criterion for families of Calabi-Yau manifolds 234.3. Some applications of the criterion 25References 38

Acknowledgement: This work is supported by The Institute of MathematicalSciences of The Chinese University of Hong Kong. I am grateful to my thesisadvisors: Shing-Tung Yau and Kang Zuo.They always encourage and support me.Special thanks to Kefeng Liu and, Andrey Todorov for their help.

Shafarevich Conjecture for Calabi-Yau Varieties 1

1. Introduction to Shafarevich Conjecture

1.1. Shafarevich conjecture over function field. At the 1962 ICM in Stock-holm, Shafarevich [26] conjectured: ”There exists only a finite number of fields ofalgebraic functions K/C of a given genus g ≥ 1, the critical prime divisors of whichbelong to a given finite set S.”

Let C be a smooth projective curve of genus g(C) over an algebraically closedfield k of characteristic 0 and S ⊂ C a finite subset. C and S will be fixed. Afamily of curves is called isotrivial,if any two general fibers are isomorphic. We canreformulate the conjecture:

Shafarevich Conjecture: Let (C, S) be fixed and q ≥ 2 an integer.

(I) There exist only finitely many isomorphism class of non-isotrivial families ofcurves of genus q over C which have at most singular fibers over S.

(II) If 2g(C) − 2 + #S ≤ 0,then there exist no such families.

In one unpublished work, Shafarevich proved his conjecture in the setting ofhyperelliptic curves.The conjecture was confirmed by Parshin for the case of S =∅,by Arakelov in general.

A deformation of a family f : X → C with the fixed base C is a family F : X →C × T such that for some t0 ∈ T ,

X Xt0 −−−−→ X⏐⏐�f ⏐⏐�FC C × t0 −−−−→ C × T

We say that two families X1 → C and X2 → C, have the same deformation type ifthey can be deformed into each other,i.e.if there exists a family X → C × T suchthat for some t1, t2 ∈ T , (Xti → C × ti) (Xi → C) for i = 1, 2.

In order to prove that there are only finitely many non-isotrivial families,one canproceed the following way.

(a) To prove that there are only finitely many deformation types,i.e.(Boundedness)

(b) To prove that the family does not admit no-trivial deformations,i.e.(Rigidity).

If (a) and (b) are right,every deformation type contains only one family and sincethere are only finitely many deformation types,this proves the original statements.

A family (f : X → C) naturally corresponds to a map ηf : C \ S → Mq,andsince C is a smooth curve,that induces a morphism ηf : C → Mq(Here Mq thecoarse moduli space for smooth curve of genus q and Mq is for stable curve.Soshown by Mumford Mq is a projective and Mq ⊂ Mq as a open subscheme) Henceparameterizing families translates to parameterizing these morphisms which can becharacterized by their graphs. The graph Γηf

of such ηf is a curve contained inC ×Mq such that the first projection maps it isomorphically onto C.Therefore theproblem is translated to look for a parametrization in the Hilbert scheme of C ×Mq.The Hilbert scheme is an infinite union of schemes of finite type,the componentscorresponding to the different Hilbert polynomials represent the deformation typesof the families. One should prove the parameterizing scheme is of finite type,i.e.onlyfinite Hilbert polynomials can actually occur.

2 Yi Zhang

In their original proofs of the conjecture,Parsin [2] and Arakelov [1] reformulatethe conjecture in the following:

A family f : X → B is called isotrivial if Xa Xb for general points a, b ∈ B(C).

Conjecture 1.1. Fixing (C, S),let q ≥ 2 be an integer.

(B) Non-isotrivial families of curves of genus q with singular locus S are param-eterized by T, a scheme of finite type .(Boundedness)

(R) All deformations of the non-isotrivial family is trivial,i.e.dim T = 0.(Rigidity)

(H) No non-isotrivial families of curves of genus q exist if 2g(C) − 2 + #S ≤0,i.e.T = ∅ ⇒ 2g(C) − 2 + #S > 0.(Hyperbolicity)

(WB) For an non-isotrivial family f : X → C,deg f∗ωmX/C is bounded above in termof g(C),#S, g(Xgen),m.In particular, the bound is independent of f .(WeakBoundedness)

In this case, because all graphs are isomorphic to C, the Hilbert polynomial isdetermined by the first term which is the deg η∗fL for a fixed ample line on Mq. Toshown boundedness is just to show deg η∗fL is bounded. Due to Mumford ’s works(one can refer theorem 3.9),it is sufficient to show (WB).Furthermore,the property(WB) will imply both (B) and (H) when the fibers are curves:

Theorem 1.2. Fixing (C, S),let q ≥ 2 be an integer.

(I) (Bedulev − Viehweg[3]) (WB) ⇒ (B).(II) (Kovacs[14] ) (WB) ⇒ (H).

One will obtains rigidity in curve case by the well known de Franchis theoremwhich is a key lemma in Faltings’s proof of Mordell Conjecture.

Exercise 1.3. f : X → C is a smooth family of curves where C is a smoothprojective curve with genus g(C) = 0 or 1.Then this family must be isotrivial.

Hint: Using the Torelli theorem which one can find in Griffiths-Harrris’s textbook.

1.2. The case of higher dimensional fibers. Let the base field is always complexnumber field C. Define Sh(C, S,K) to be the set of all equivalent classes of non-isotrivial family {f : X → C} such that Xb is a smooth projective variety with ’type’K for any b ∈ C \ S. Two such families are equivalent if they are isomorphic overC − S.

The general Shafarevich type problem is:

For which type K of varieties and data (C, S),is Sh(C, S,K) finite ?

Example 1.4. [8]Faltings has dealt with the case where the fibers are Abelianvarieties,and he formulates a Hodge theoretic condition for the fiber space to berigid. This condition is always called Deligne-Faltings (∗) condition:

A smooth family f : X0 → C \ S of Abelian variety satisfies (∗) if any anti-symmetric endomorphism σ of VZ = R1f∗Z defines an endomorphism of X0(so σ isof type (0, 0)).

Let Q be the sympletic bilinear form on R1f∗Z.By global Torelli theorem Q isinduced by a polarization of the Abelian scheme.It is shown by Faltings that the

Shafarevich Conjecture for Calabi-Yau Varieties 3

(∗)-condition is equivalent to

EndQ(V) ⊗ C = (EndQ(V) ⊗ C)0,0.

On the other hand,the Zariski tangent space of the moduli space of the Abelianschemes over C − S with a fixed polarization type is isomorphism to

(EndQ(V) ⊗ C)−1,1.

Thus one obtains the rigidity.

Example 1.5. [24] If a family is not rigid,then the corresponding VHS is surelynonrigid. By utilizing differential geometric aspects of the period map and theassociated metrics on the period domain,Peters extend Faltings’s result to generalpolarized variation of Hodge Structure of arbitrary weight

A polarized variation of Hodge Structure underlying VZ is rigid if and only if(EndQ(VQ) ⊗ C)−1,1 = 0.

Example 1.6 (Jost-Yau). Using techniques from harmonic maps Jost and Yauanalyzed Sh(C,E,Z) for a large class of varieties. See [10]. They gave differentialgeometric proofs of above theorems.Their paper provided analytic methods to solverigidity problems, and also gave a powerful tool to analyze the Higgs bundles withsingular Hermitian metric.

The deformation of the family can be reduced to deformation of the correspondingperiod map.It is interesting to study the case that the period domain is Hermitiansymmetric space.

Example 1.7 (Mok Ngaiming). Considering those arithmetic varieties arising asmoduli spaces for certain polarized Abelian varieties,Mok [20] proves a finitenesstheorem for the Mordell-Weil group (i.e., the group of holomorphic sections) of theassociated universal Abelian variety.

Let Ω/Γ be a quotient of a bounded symmetric domain by a discrete properly dis-continuous subgroup Γ in Aut(Ω), Mok and Eyssidieux show that for any immersedcompact complex submanifold S ↪→ Ω/Γ, assume that the tangent subspaces aregeneric in some algebro-geometric sense,S can be locally approximated by a uniqueisomorphism class of totally geodesic complex submanifolds of Ω.

Example 1.8 (No rigid families of Abelian varieties). Faltings constructed examplesshowing that Sh(C,E,Z) is infinite for Abelian varieties of dimension ≥ 8. See [8].Saito and Zucker extended the construction of Faltings to the setting when Z is analgebraic polarized K3 surface. They were able to classify all cases when the setSh(C,E,Z) is infinite.

For boundedness of above examples,one may refer to the works of [8],[22],[?].It should be pointed out they were not considering polarized families.In the case of fibers are curve, we assume the condition q = g(Xgen) ≥ 2 which is

equivalent to the condition that the canonical line bundle of generic fiber KXgen isample.

The role of the genus is played by the Hilbert polynomial, thus fixing g(Xgen) canbe replaced by fixing hKXgen

, the Hilbert polynomial of KXgen . And over C − S,thefamily f : X → C should be smooth canonical polarized family .

4 Yi Zhang

In higher dimension case,instead of fixing g(Xgen), it is reasonable to we thinkabout the polarized projective variety (X,L) such that L is ample line bundle on Xand χ(X,Lν) is a fixed Hilbert polynomial h(ν).

As pointed out in following section,by Matsusaka-Mumford Big theorem L canbe regard as a very ample line bundle such that N = h(1) − 1 is independent onL,there is an embedding

j(X) : X ↪→ PN .

Let C be a fixed nonsingular projective curve and S be fixed finite points on C.Letf : X → C be a family which is asked to satisfied the following condition:

Over C \S, (f : (X ,L) → C \S) is a polarized family such that Lb = L|Xbample

on Xb andχ(Xb,Lνb ) = h(ν)

the fixed Hilbert polynomial for any b ∈ C \ S.

With the polarization condition, we define Sh(C, S,K) to be the set of all equiv-alent classes of non-isotrivial family {f : X → C} as above such that Xb is a smoothprojective variety with ’type’ K for any b ∈ C \S. Two such families are equivalentif they are isomorphic over C − S as polarized families.

One can formulate the problem into:

Conjecture 1.9 (Higher dimensional Shafarevich conjecture ). Fixing (C, S)and the Hilbert polynomial h(ν).

(B) The elements of Sh(C, S,K) are parameterized by T,a scheme of finite typeover C.

(R) dim T = 0.(H) T = ∅ ⇒ 2g(C) − 2 + #S > 0

The important works of Viehweg on Moduli problem together with so called Arakelov-Yau inequality guarantee the Boundedness of Analogue Shafarevich conjec-ture for the family of polarized of Calabi-Yau manifolds.

One also can refer to the paper [18] for the proof by using the well known Schwarz-Yau lemma and Bishop compactness.

In higher dimensional fibers case,one also have the weak boundedness conjecture:

(WB) For a family (f : X → C) ∈ Sh(C, S,K),deg f∗ωmX/C is bounded above interm of g(C),#S, h,m.In particular, the bound is independent of f .

These theorem and proposition together with so called Arakelov-Yau-Schwarzinequality guarantee the Boundedness of Analogue Shafarevich conjecture for thefamily of polarized of Calabi-Yau manifolds.

(WB) ⇒ (H) is still true for the canonically polarized families.But it is an enigmawhether (WB) implies (B) in higher dimensional fibers case.

Example 1.10. (New important results).

(M-K-Z) Migliorni,Kovac and Zhang Qi proved that any family of minimal algebraicsurfaces of general type over a curve of genus g and m singular points suchthat 2g(C)− 2+#S ≤ 0 is isotrivial. See [13], [19], [40] and [3]. Oguiso andViehweg[23] proved the same results for families of elliptic surfaces.

Shafarevich Conjecture for Calabi-Yau Varieties 5

(B-V) Bedulev and Viehweg [3] have proved the boundedness for families of alge-braic surfaces of general type over a fixed algebraic curve, the weak bound-edness for family of canonically polarized varieties.

(V-Z) Recently Viehweg and Zuo have obtained very important results [32][33] [34]:Brody hyperbolicity was proved for the moduli space of canonically polar-ized complex manifolds. They proved the boundedness for Sh(C,E,Z) forarbitrary Z, with ωZ semi-ample. They also established that the automor-phism group of moduli stacks of polarized manifolds is finite. The rigidityproperty for the generic family of polarized manifolds has been proved too.

(L-T-Y-Z) In their preprint“The Analogue of Shafarevich’s Conjecture for Some CYManifolds”, Liu,K.,Todorov,A.,Yau,S.T.,Zuo,K give a simple and readableproof of the boundednes( Their idea is to use Schwarz-Yau lemma and Bishopcompactness).They spend most chapters to deal with rigidity by the idea ofusing Yukawa coupling.

Remark: A complex analytic space N is called Brody hyperbolic if every holomor-phic map C → N is constant.

Certainly,we have an algebraic version:algebraic hyperbolic.The essential fact isthat if the moduli space is algebraic hyperbolic,then (H) in Shafarevich conjecturewill hold. It is the motivation for us to study the Brody hyperbolic.

Here are the new process on the conjecture:

Notations 1.11 (Key Observation by Viehweg and Zuo[36]). The rigid prop-erty for the generic family of polarized manifolds has been proved by Viehweg andZuo. Now they point out that the conjecture will fail for general condition thoughmost families of Calabi-Yau manifolds are rigid. They construct some importantcounterexamples and show a principle that there always exist a product of the mod-uli space of hypersurface of degree d in Pn embedded into the moduli space ofhypersurface of degree d in PN where N > n.

So the first key step for Shafarevich conjecture is to find a morefine condition.

Question : Can we find the necessary and sufficient condition? It is alsointeresting to classify the non-rigid families of Calabi-Yau manifolds.

Example 1.12 (Zhang [41][42]). In his Thesis,Zhang proves(1) Lefschetz pencils of Calabi-Yau manifolds of odd dimension are rigid.The

proof depends on the special properties of Lefschetz pencils. As Deligneshowed in his proof of Weil conjecture I,the VHS of Lefschetz pencils canbe decomposed into two sub VHSs: One is invariant space and anotheris vanishing cycles space which is absolute irreducible under the action offundamental group π1(P1\S) where S is set of singular values(this is essentialthe Kazdan-Magulis theorem).

According to this observation, Zhang shows that for arbitrary given non-trivial pencil of CY manifolds,the vanishing cycles space is not empty andthe pieces of (n, 0)-type and (0, n)-type of the VHS are both in the vanishingcycles space. But on the other hand,assuming the family is not rigid,one canobtain a flat non zero (−1, 1) type endomorphism σ of the VHS. It makes

6 Yi Zhang

the local system splitting such that (n, 0) and (0, n) are in different factorsby local Torelli theorem. Therefore,it is a contradiction.

(2) Zhang obtains a general result on so called strong degenerate families( notonly for families of Calabi-Yau manifolds)which certainly are of ”gen-eral” type.We call a family over any closed Riemann surface strong degen-erate if it has a singular fiber such that every component is dominated byprojective space Pn. The author shows this family must be rigid. He usesthe similar trick in (1) to get a nonzero endomorphism σ.By the propertiesof Higgs bundles,he identifies this σ to be a monodromy-invariant sectionof the VHS of the self-product family by Kunneth formula.Finally, usingthe properties of the strong degenerate singular fiber,he shows the endomor-phism σ is zero so that the family must be rigid.As a corollary,he obtains aweak Arakelov theorem of high dimensional version.

(3) Following the recent works of Liu-Todorov-Yau-Zuo and Viehweg-Zuo,Zhang gives another proof of a criterion of rigidity by using the technique ofHiggs bundles: A Calabi-Yau family with nonzero Yukawa coupling shouldbe rigid. As an application of this criterion,with the arguments of Schmidand Simpson on the residues of vector bundles over singularizes,the authorshows that families of CY manifolds admitting a degeneration with maximalunipotent monodromy must be rigid.

1.3. Appendix: Arithmetic version of Shafarevich conjecture. Let (R,m)be a DVR, F = Frac(R),and C a smooth projective curve over F .C is said to havegood reduction over R if there exists a smooth projective variety B over Spec(R)such that

C�−−−−→ BF −−−−→ B⏐⏐� ⏐⏐�

Spec(F ) −−−−→ Spec(R)

.

Therefore,let R be a Dedekind ring,F = Frac(R),and C a smooth projective curveover F . C has good reduction at the closed point m ∈ Spec(R) if it has a goodreduction over Rm.

Conjecture 1.13 (Shafarevich). Let q ≥ 2 be an integer,F a number field,R ⊂ F

the ring of integers of F ,and Δ ⊂ SpecR a finite set.Then there exists only finitelymany smooth projective curves over F of genus q that have good reduction outsideΔ.

The Shafarevich’s conjecture was inspired by the well-known Hermit theorem inalgebraic number theory :

the number of extensions k′/k of a given degree whose critical prime divisorsbelong to a given finite set S is finite.

Therefore,we can get the function field version of the conjecture is :Let q ≥ 2 be an integer,F = K(B) the function field of B,R the subring of F such

that B − Δ = Spec(R) a number field,R ⊂ F the ring of integers of F .Then thereexists only finitely many smooth projective non-isotrivial curves over F of genus qthat have good reduction over all closed points of Spec(R).

Shafarevich Conjecture for Calabi-Yau Varieties 7

The arithmetic version was confirmed by Faltings in 80s and he used it to provethe

Conjecture 1.14 (Mordell). Let F be a number field and C a smooth projectivecurve of genus q ≥ 2 defined over F . Then C(F ) is finite.

8 Yi Zhang

2. The Modern Proof of Original Shafarevich Conjecture

Here we will give a sharp proof due to the Viehweg and Zuo.

2.1. Moduli of Curves. Recall that a reduced projective curve C is called stable, ifthe singularities of C are normal crossings(it is locally isomorphic to the singularityof the plane curve xy = 0), and if ωC is ample.

Theorem 2.1 ((Mumford [21])). For g ≥ 2, define

Mg(C) = { stable curves of genus g, defined over C}/ ∼= .

Then there exists a projective coarse moduli scheme Mg for Mg, of dimension 3g−3.i.e. a variety Mg and a natural bijection Mg(C) ∼= Mg(C) where Mg(C) denotesthe C-valued points of Mg.

LetMg(C) = { smooth curves of genus g, defined over C}/ ∼= .

Similar, there is a quasi-projective coarse moduli scheme Mg for Mg and Mg ⊂ Mg

as Zariski open set.

Remarks :

1. Stable curve means Deligne-Mumford stable curve,the definition is equivalent toa reduced irreducible projective curve with only ordinary doubles as singularitiesand only finitely many automorphisms. The restriction on the number of auto-morphisms thus means that every smooth rational component of a stable curveintersects the remaining components in at least three points.

2. The moduli scheme Mg is normal, connected and reduced.The subscheme Mg,corresponding to non-singular curves of genus g, is open in Mg. As in GIT,onecan use “level μ-structures”to shown that the moduli schemes Mg have finitecoverings φ : MΓ

g → Mg which carry a universal family

g : XΓ → MΓg .

3. We will give the precise definition of a coarse moduli scheme late. Let us justexplain that the “natural” meaning of coarse moduli space :

For each flat family g : X → Z, whose fibers g−1(z) belong to Mg(C), theinduced map Z(C) → Mg(C) comes from a morphism of schemes φ : Z →Mg.(Hence for every family (g : X → Z) of stable curves of genus of g,thereexists a morphism ηg : Z → Mg such that for all z ∈ Z,ηg(z) = [Xz].)

It follows from the construction of Mg, that for all ν > 0 and for some p � ν

there exists an invertible sheaf λ(p)ν , such that for all families g : X → Z,

det(g∗ωνX/Z)p = φ∗(λ(p)ν ).

Proposition 2.2 ((Mumford [21])). For ν, μ and p sufficiently large and divisible,for

α = (2g − 2) · ν − (g − 1) and β = (2g − 2) · ν · μ− (g − 1)

the sheaf λ(p)α

ν·μ ⊗ λ(p)−β·μν is ample.

Shafarevich Conjecture for Calabi-Yau Varieties 9

Definition 2.3. E be a locally free sheaf on a curve Y ,We say E numerically effec-tive(nef) if OP(E)(1) is nef line bundle on projective space P(E).Similiarly we get thedefinition of ample vector bundle.

Remark: [31]Let H be an ample line bundle on Y ,U Zariski open set of Y .

• It is not difficult to show that E is nef if and only if for all finite covering π : C → Y

and for all invertible quotients π∗E � N ,one has degN ≥ 0.• It is shown by Hartshone that E is ample(with respect to U) if and only if for

some η > 0 there exists a morphism⊕H −−→ Sη(E),

surjective (over U).• It is natural that we have the definition weakly positive:

E is weakly positive over U ,if for all α > 0 the sheaf Sα(E) ⊗ H is ample withrespect to U .

• One can use this properties to give similar definition on a quasi-projective varietyZ.Ther is a little modification: E is called nef on Z, if for all morphisms π : C →Z, from a curve C to Z, and for all invertible quotients π∗E � N one hasdeg N ≥ 0.

Exercise 2.4. Given d ∈ N, assume that for all δ ∈ N−{0}, there exists a coveringτ : Y ′ → Y of degree δ such that τ∗E ⊗H is nef, for one H of degree d(hence for allinvertible sheaves of degree d). Then E is nef.

2.2. Rigidity property.

Conjecture 2.5 (Viehweg). On U = Z − S, for ϕ : U → Mg induced by a family

(a) ϕ generically finite =⇒? Ω1Z(logS) weakly positive?

(b) ϕ generically finite =⇒? det(Ω1Z(logS)) big ?

Viehweg-Zuo’s Theorem[33] For η � 1 there is an invertible subsheaf

L ⊂ Sη(Ω1Z(logS))

with κ(L) ≥ dim(ϕ(Z)). Therefore,one has Conj.(a) =⇒ Conj.(b).

Theorem 2.6 (Zuo[44]). Conjecture (a) holds true if the fibres of X0 → U satisfylocal Torelli theorem.

Because local Torelli theorem always holds for curve case,the conjecture (a) holdsfor families of algebraic curves(Originally,this fact is bellowed to Mumford in hisworks on Moduli space of curves.)

Corollary 2.7. Zuo’s theorem implies the Rigidity property of the Shafarevichconjecture.

Proof. Assume a non-isotrivial family over curve Y is not rigid then there existscurve T0 and smooth extended family of curves

X0 −−→ Z0 = T0 × Y0,

and the induced morphism φ : Z0 →Mg is generically finite.

10 Yi Zhang

In fact, it is impossible by Zuo’s theorem. Otherwise, replacing Y0 and T0 bysome covering, one would find for compactifications T and Y of T0 and Y0 the sheaf

Ω1T×Y (log (T × (Y − Y0) + (T − T0) × Y )) =

pr∗1Ω1T (log (T − T0)) ⊕ pr∗2Ω

1Y (log (Y − Y0))

to be ample over some dense open subset. Obviously this cannot be true: Choose ageneric quasi-projective curve Y0 × t,still denote it Y0, thenΩ1T×Y (log (T ×(Y −Y0)+(T −T0)×Y ))|Y0 will be ample over Y0.But it is impossible

because(pr∗1Ω

1T (log (T − T0))

⊕�)|Y0 = ⊕OY0

⊕�|Y0 .

�Remarks :1. That the canonical morphism is generically finite is equivalent to that there are

no isotrivial subfamilies including a general fiber.2. Using the argument in Chapter 4, we will have a nonzero section

OT0 → φ∗(Ω∨T0

),

then we get a contradiction.

2.3. Boundedness. Here I would like to give a proof by using the arguments ofKefeng Liu in [15],[16].

2.3.1. Teichmuler space Tg and period map. Fix a compact Riemann surface R0 ofgenus g.Consider a pair (R,H) of a compact Riemann surface R and a homotopyclass H of orientation-preserving homeomorphisms from R0 to R.Define an equiva-lence relationsim:

(R,H) ∼ (R′, H ′) ⇐⇒ H ′ ◦H contains a biholomorphic map from R to R′.

Let Tg := {(R,H)}/ ∼,the classic Teichmuler theorem tell us that Tg has a complexstructure and is a smooth complex ball. Fix a symplectic basic

{α1, · · · , αg, β1, · · · , βg}of H1(R0,Z).Then,for (R,H) the set

{H(α1), · · · , H(αg), H(β1), · · · , H(βg)}is a symplectic basis of H1(R,Z).We choose a basis {ω1, · · · , ωg} of holomorphic1-form on R such that ∫

H(βi)ωj = δij , 1 ≤ i, j ≤ g.

Then these integral determine uniquely on the equivalence class a point

(∫H(αi)

ωj)

of Siegel upper half space Dg. Exactly,there is a holomorphic map

τ : Tg −→ Dg,

[(R,H)] �−→ (∫H(α)

ωj).

Shafarevich Conjecture for Calabi-Yau Varieties 11

The standard Teichmuler theory tell us there is a modular group Modg which is asubgroup of Sp(2g,Z) such that

Mg = Modg\Tg.Thus τ descends to a holomorphic map

j : Mg → Ag = Sp(2g,Z)\Dg,

and dimC Ag = g(g + 1)/2.Geometry on Tg : For a compact Riemann surface of genus g ≥ 2,we know

H2(R,ΘR) = 0 and dimCH1(R,ΘR) = 3g − 3.This implies that the base space

T of the Kuranishi family U → T is a smooth complex manifolds with dimension3g−3 and the Kuranishi family is universal at each point of T because H0(R,ΘR) =0.Moreover,there is a complex analytic family

π : R → Tgof the compact Riemann surface of genus g and that it is universal at every point ofTg.However there is no analytic family over Mg,this is why we call it coarse modulispace.We haveTorelli Theorem

(1) τ : Tg → Dg is locally injective,(2) j : Mg → Ag = Sp(2g,Z)\Dg is injective.

2.3.2. Weil-Petersson metric. Let π : X → Tg be the universal family over theTeichmuller space. The Poincare metric on each fiber patches together to give asmooth metric on ΩX/Tg

. Then π∗Ω⊗2X/Tg

, the push-down of Ω⊗2X/Tg

, is the cotangentbundle of Tg. Recall that for any point s ∈ Tg

R0π∗Ω⊗2X/Tg

|s = H0(Xs,Ω⊗2Xs

) ⊗ k(s)

where Xs = π−1(s).There exists a natural inner product on π∗Ω⊗2

X/Tginduced from the Poincare metric

on each fiber.GWP (s)(μ1, μ2) =

∫Xs

< μ1, μ2 >ρ(s)

where μ1, μ2 ∈ H0(Xs,Ω⊗2Xs

) and ρ(s) is Poincare metric on Xs. This inner productinduces the Weil-Petersson metric on π∗Ω⊗2

X/Tg.Moreover,shown by Wolpert [37]

(2.7.1) π∗c21(ωX/Tg

) =1

2π2ωWP.

where ωWP is the Kahler form of GWP .So Weil-Petersson metric is Kahler met-ric,furthermore it is not complete.

Properties 2.8. Important facts:

(1) Shown by Alforhs the sectional curvature of GWP is bounded from above by

− 12π(g − 1)

.

(2) There is a obvious fact that the Weil-Petersson metric is invariant underthe modular group Modg,thus the equation 2.7.1 still hold for the universalfamily over Mlevel

g .

12 Yi Zhang

(3) It can be shown that,as currents the equation 2.7.1 holds on Mlevelg .

2.3.3. Weak Boundedness. Given a family f : X → Y over projective curve,wealways assume it is stable family(!we can do it) and the restricted family f : X0 →Y0 = Y − S is smooth. Moreover we can assume the image of moduli map is in thefine moduli space Mlevel

g ) with level structure(we can get it by Galois covering).Thefamily is uniquely induced from the universal family U → Mlevel

g ,then the Weil-Petersson metric can descend to Mlevel

g .Remark: Mlevel

g will be of Γ\Tg where Γ is subgroup of Sp(2g,Z) with a levelstructure.

We have the moduli map g : Y0 → Mlevelg and

(f : X0 → Y0) ∼= U ×g Y0 → Y0.

we have the intersection(Calculated by Liu,K.F in [15] [16]).

(ω2X/Y ) =

∫Xc21(ωX/Y ) =

∫Yf∗c

21(ωX/Y )

By the current property in 2.8,∫Yf∗(c21(ωX/Y )) =

∫Y0

f∗c21(ωX0/Y0

) =1

2π2

∫Y0

g∗ωWP .

But ∫Y0

g∗ωWP < 2π(g − 1)∫Y0

ωP = 2π(g − 1)(2g(Y ) − 2 + #S)

where ωP is the Poincare metric on Y0.The last step is Gauss-Bonnet and the in-equality is the following :

Theorem 2.9 (Schwarz-Yau Lemma). Let R be a compact Riemann surface (g >1)with curvature −1 or an affine Riemann surface of hyperbolic.Let N be Hermitainmanifolds with holomorphic sectional curvature strictly bounded above by negativeconstant −K.Then for any non-constant holomorphic map f from R to N ,one has

f∗ωN <1KωR

where ωR, ωN denote respectively the Kahler form of R,N .

Using a inequality of Xiao Gang

deg f∗ωX/Y ≤ (g

4g − 4)(ω2

X/Y )

we get (WB).By the theorem of Bedulev-Viehweg [3],(WB) ⇒ (B).

Remark:Exactly,Deligne proved that

deg f∗ωX/Y ≤ g

2(2g(Y ) − 2 + #S).

Jost and Zuo have a smart proof the weak boundedness.Moreever,Zuo has a se-ries works on the powerful Arakelove-Yau inequality in case of higher dimensionalfiber. Using Miyaoka-Yau inequality,Tan S.L proved the Beauville’s conjecture(Inthe papers of Liu which I mention above,he also solved the problem by the methodI show),then got a more fine estimate: for semistable fiberation with g ≥ 2

deg f∗ωX/Y <g

2(2g(Y ) − 2 + #S).

Shafarevich Conjecture for Calabi-Yau Varieties 13

3. Moduli Properties of Families of Calabi-Yau Manifolds

3.1. Family of polarized algebraic manifolds. In last chapter of the thesis,theauthor will study the rigid problem of the family of polarized variety. Why shouldwe study the polarized family ?

Generally,for technical reason, we should endow Moduli stack of the geometricobject with the given type a nice algebraic structure such that every family of thegiven type is induced geometrically from the it(Universal property). In fact, in mostcases the fine moduli scheme (which other family is uniquely induced from) does notexist.Fortunately,Viehweg has shown that the coarse moduli space of polarized vari-eties with semi-simple canonical line bundle and given Hilbert polynomial exists andis quasi-projective. As an application,one obtains the quasi-projective moduli spacefor K3 surfaces,Abelian varieties and Calabi-Yau manifolds.With these results andrecent works of Zuo Kang on Arakelov-Yau’s inequalities,one can obtain the bound-edness of the Shafarevich conjecture for Calabi-Yau manifolds. On the shoulderof Viehweg and Zuo’s working, it is meaningful to consider the rigid problem ofShafarevich conjecture.

Now, Let the ground field be complex field C.

Definition 3.1. (Moduli Problem [31])

1. The objects of a moduli problem of polarized schemes will be a class F(C),consisting of isomorphism classes of certain pairs (W,H), with:a) W is a connected equidimensional projective scheme over C.b) H is an ample invertible sheaf on Γ or, as we will say, a polarization of W

2. For a scheme Y a family of objects in F(C) will be a pair (f : X → Y,L) whichsatisfiesa) f is a flat proper morphism of schemes,b) L is invertible on X,c) (f−1(y),L|f−1(y)) ∈ F(C), for all y ∈ Y ,d) some additional properties, depending on the moduli problem one is interested

in.3. If (f : X → Y,L) and (f ′ : X ′ → Y,L′) are two families of objects in F(C) we

write (f,L) ∼ (f ′,L′) if there exists a Y -isomorphism τ : X → X ′, an invertiblesheaf F on Y and an isomorphism τ∗L′ ∼= L ⊗ f∗F . If one has X = X ′ andf = f ′ one writes L ∼ L′ if L′ ∼= L ⊗ f∗F .

4. If Y is a scheme over C we define the Moduli Functor

F : (Sch/C)o −−→ (Sets)

by

F(Y ) = {(f : X → Y,L); (f,L) a family of objects in F(C)}/ ∼ .

The natural functor transform for any τ : Y ′ −−→ Y with Y, Y ′ ∈ (Sch/C) is

F(τ) : F(Y ) −−→ F(Y ′)

by sending (f : X → Y,L) in F(Y ) to (pr2 : X ×Y Y′ → Y ′, pr∗1L) ∈ F(Y ′).

Let us introduce a coarser equivalence relation on F(C) = F(Spec(C)) and onF(Y ), which sometimes replaces “∼”.

14 Yi Zhang

Definition 3.2. Let (f : X → Y,L) and (f : X ′ → Y,L′) be elements of F(Y ).Then (f,L) ≡ (f ′,L′) if there exists an Y -isomorphism τ : X → X ′ such thatthe sheaves L|f−1(y) and τ∗L′|f−1(y) are numerically equivalent for all y ∈ Y . Bydefinition this means that for all curves C in X, for which f(C) is a point, one hasdeg(L ⊗ τ∗L′−1|C) = 0.

Then one defines functors PF from the category of C-schemes to the category ofsets by choosing

1. On objects: For a scheme Y defined over C one takes for PF(Y ) = F(Y )/ ≡ theset defined as in 3.1, 4).

2. On morphisms: For τ : Y ′ → Y one defines

PF(τ) : PF(Y ) → PF(Y ′)

as the map obtained by pullback of families.

We will call F the moduli functor of the moduli problem F(C) and PF the modulifunctor of polarized schemes in F(C), up to numerical equivalence.

If F′(C) is a subset of F(C) for some moduli functor F then one obtains a newfunctor by choosing

F′(Y ) = {(f : X → Y,L) ∈ F(Y ); f−1(y) ∈ F′(C) for all y ∈ Y }.

We will call F′ a sub-moduli functor of F.

In order to make sense of the definition of 3.1,one should make precise the pairsin 1) and the additional properties in 2).Thus,we have

Definition 3.3. (Moduli problem on polarized manifolds)

1. Polarized manifolds: M′ is the moduli functor such that

M′(C) = {(W,H); W a projective manifold, H ample invertible on W}/ ∼

and defines again M′(Y ) to be the set of pairs (f : X → Y,L), with f a flatmorphism and with L an invertible sheaf on X, whose fibres all belong to M′(C).

2. Polarized manifolds with a semi-ample canonical sheaf:M is the moduli functor given by

M(C) = {(W,H); Γ a projective manifold, H ampleinvertible and ωW semi-ample }/ ∼

and, for a scheme Y , by defining M(Y ) to be the subset of M′(Y ), consisting ofpairs (f : X → Y,L), whose fibres are all in M(C). We write P instead of PM

for the moduli functor, up to numerical equivalence.

Remark 3.4. Let h(T ) ∈ Q[T ] be a polynomial with h(Z) ⊂ Z,then for the modulifunctor F as one of above functors,one defines Fh by

Fh(Y ) = {(f : X → Y,L) ∈ F(Y );h(ν) = χ(Lν |f−1(y))for all ν and all y ∈ Y }.

Thus for (Γ,H) ∈ F(C),h(T ) is the Hilbert polynomial of H.Moreover,

F(Y ) =•⋃h

Fh(Y ).

Shafarevich Conjecture for Calabi-Yau Varieties 15

If the Moduli functor can be represented(i.e F = Hom(−,M) for some schemeM), then we say the Moduli scheme exists.Geometrically,it just means whether theuniversal family exists.

Definition 3.5 (Moduli Space). Mumford’s definition in GIT [21]

(A) A fine moduli scheme for F consist of a scheme M and a universal family(g : X →M,L) ∈ F(M).Here universal means,that for all (f : X → Y,H) ∈F(Y ),there is a unique morphism τ : Y → M with

(f : X → Y,H) ∼= F(τ)(g,L) = (X ×M Ypr2−−→ Y, pr∗1L).

(B) A coarse moduli scheme for F is a scheme M together with a bijectionμ : F(C) ↔ M(C) such that for any family (f : X → Y,L) ∈ F(Y ):(1) The induced map of set φ : Y (C) → M(C) comes from a morphism

Y → M.

(1) Given a scheme N/C and a natural transformation

Φ : F → Hom(−, N),

the map of set Φ(C) ◦ μ−1 : M(C) → N(C) comes form a morphism.

Exercise 3.6. It is easy to check the coarse moduli space is unique in the followingmeaning: For any two coarse moduli space (Mi, μi) i = 1, 2 of the functor F, thereis an isomorphism Ψ : M1 → M2 such that Ψ ◦ μ1 = μ2.

Theorem 3.7 (Viehweg,E.[31]). The Existence of Coarse Moduli

1. Given a polynomial h ∈ Q[T1, T2] of degree n such that with h(Z × Z) ⊂ Z,Then,there exists a coarse quasi-projective moduli scheme Mh for the sub-modulifunctor Mh(same for PMh),of polarized manifolds (W,H) ∈ M(C),with

h(α, β) = χ(Hα ⊗ ωβW ) for allα, β ∈ N

Furthermore,the moduli functor Mh(same for Ph) is bounded by Matsusaka Bigtheorem.

2. Given a polynomial h of degree n such that h(Z) ⊂ Z,there exists a quasi-projective scheme Mh of finite type over C for the set

{(X,L);ωX semi-ample,L ample on X and χ(X,Lν) = h(ν)for all ν}/ ∼ .

Furthermore,Mh is bounded by Matsusaka Big theorem.As in item (I), the quasi-projective coarse moduli space for PMh exists and is

bounded too.

Here L semi-ample means that if for some μ > 0 the sheaf Lμ is generated by globalsections.

Corollary 3.8. In particular,adding the condition as canonical line bundle trivial(the deformation invariant condition),one may get the coarse quasi-projective modulispace for K3 surfaces, Calabi-Yau manifolds and Abelian variety.Also these Modulifunctor are bounded.

Proposition 3.9 (Mumford-Viehweg [31]). Let the family (g : X → Y ) be inducedby the morphism φ : Y → Mh.

16 Yi Zhang

(I) Assume all Γ ∈ Mh(C) are canonical polarization manifolds.For η ≥ 2 withh(η) > 0,there exists some p > o and an ample invertible sheaf λ(p)

η on Mh

such thatφ∗λ(p)

η = det(g∗ωηX/Y )p

(II) Assume for some δ > 0,one has ωδΓ = OΓ all manifolds Γ ∈ Mh(C)(for example,K3 surfaces,Calabi-Yau manifolds,Abelain varieties,etc). Thereexists some p > o and an ample invertible sheaf λ(p)

η on Mh such that

φ∗λ(p) = g∗ωδ·pX/Y

Theorem 3.10 (Matsusaka-Mumford Big Theorem). Assume the ground field isalways zero.Let M be the set of isomorphism of polarized smooth projective varietieswith a fixed Hilbert polynomial h. Then M is bounded,i.e, ∃ an integer m0 indepen-dent of the choice of (X,H) ∈ M such that m0H is very ample for m ≥ m0 on X andin fact one can also suppose that Hj(X,mH) = 0 for j > 0 and m ≥ m0.Hence,thecomplete linear system |m0H| gives a closed immersion:

i : X ↪→ PN

with N = h(m0)−1.In particular the Hilbert polynomial of i(X) is h1(m) = h(mm0)and hence independent of (X,H) ∈ M.

Therefore, geometrically,the families we study are

f : X →M

where X and M are algebraic manifolds defined over C and f is surjective morphismwith following conditions:(a) f is a smooth morphism with every closed fiber f−1(t) a connected and reducedsmooth projective variety.(b) f is a projective morphism i.e. there exists a projective space PN such that thediagram is commutative:

Xi−−−→ PN ×M

���

f ���

π

M(c) the varieties X and M and every closed fiber Xt = f−1(t) are connected. More-over,let ωt = OPN (1)|Xt then (Xt, ωt) is a polarized algebraic variety such that theembedding it : Xt ↪→ PN is determined by the very ample line bundle ωt,thus ωt isalso a Kahler structure of Xt.

Remark: We can assume the conditions (b) and (c) according to the Matsusaka-Mumford Big theorem.Because of the flatness of f ,the dimension of the closed fiberis a constant.

3.2. Deformation theory and Torelli theorem.

3.2.1. Deformation theory. Let f : X →M be a smooth family of complex manifoldsover a complex manifold M ,Let t0 ∈M such that X = Xt0 = f−1(s0).Such a familyis called a deformation of the complex manifold X.

There exists an exact sequence of OX-homomorphisms

0 −−→ TX/M −−→ TX −−→ f∗TM −−→ 0

Shafarevich Conjecture for Calabi-Yau Varieties 17

Form the exact sequence one obtains a long exact sequence

· · · −−→ f∗TX −−→ f∗f∗TM = TM

κ−−→ R1f∗TX/M −−→ R1f∗TX −−→ · · ·Then the OM -homomorphism

κ : TM −−→ R1f∗TX/M

is just the Kodaira-Spencer map,i.e. κ is in the class H0(M,R1f∗TX/M ⊗ Ω1M ).At

any point t ∈M ,κ(t) : TM,t −−→ H1(Xt, TXt).

Let M be the moduli space of the polarized algebraic variety (X,L),while D isthe classifying space of the polarized Hodge Structure of weight n associated to(X,L),there is a natural mapping

φ : M → D/GZ

which is an extended variation of Hodge structures.Intuitionally,the Kuranishi base (refer to the Kodaira’s textbook)is the maximal

deformation space of X which is contained in Euclidean space H1(X,ΘX). In gen-eralization, The Kuranishi family always exists for each complex manifold X, and ifdimCH

0(Xs,ΘXs) keeps constant in the neighborhood of t0,the family is universal(refer to Kuranishi’s works). Given a orientation η ∈ H1(X,ΘX),whether X can bedeformed along η is dependent on the corresponding [η, η] ∈ H2(X,ΘX). The set ofall [η, η] inH2(X,ΘX) is called obstruction class.Thus if the obstruction class van-ish,the Kuranish base is a smooth complex manifold of dimension dimCH

1(X,ΘX).In particular,for the example when H2(X,ΘX) = 0.

Bogomolov,Todorov,Tian, Ran and Kawamata proved that

Theorem 3.11 (BTT). Let X be a compact Kahler manifold with trivial canon-ical bundle,then the Kuranishi family is universal, and the base is a smooth openset of dimension dimCH

1(X,ΘX). In particular,the result holds for X Calabi-Yaumanifold.

Remark 3.12. Actually, the family is universal at every point of the base and K-Smapping is an isomorphism at every point of the base.

3.2.2. Torelli theorem. What’s the meaning of Torelli theorem? Roughly speck-ing,the question is equivalent to that for a compact Kahler manifold, whether theHodge Structure determines the complex structure.

Suppose a compact Kahler manifold X admits a universal Kuranishi family

f : (X, X) → (S, 0)

with a nonsingular base S which is isomorphic to the open set in H1(X,ΘX) .Certainly,this may not be polarized family. There is a well-defined holomorphicmap (period map)

λ =∏

λp : S →∏

Gr(hp, H)

The infinitesimal Torelli theorem is just to ask whether (dλ)0 is injective.Nowone think about the Kuranishi family of the polarized variety (X,L) where X isprojective manifold and L an ample very bundle.

18 Yi Zhang

Consider the submanifold Sc1(L) ⊂ S,the deformation space determined by c1(L),the restriction of f over Sc1(L) is the universal Kuranishi family of the polarizedalgebraic variety (X,L) ( i.e any deformation φ : (Y, X) → (T, 0) of polarized variety(X,L) is locally obtained from f by a unique base change π : (T, 0) → (Sc1(L), 0)).We also have

TSc1(L),0 = H1(X,Θ)c1(L) := Ker(H1(X,ΘX)∧c1(L)−−−−→ H2(X,OX))

Suppose Φ is the period mapping for f over Sc1(L) and ΦT is period mapping forT ,Thus the diagram is commutative(locally,then in this case Γ = {1} )

Tπ−−→ Sc1(L)

���

ΦT�

��Φ

D

It is said that infinitesimal Torelli theorem holds for a polarized algebraic variety(X,L) if Φ is local embedding. The condition is equivalent to that dΦ and dλ isinjective on holomorphic tangent space ΘSc1(L),0.

On the other way,the bilinear ΘX × Ωn−1X defines a paring

H1(X,ΘX) ×Hn−p,p → Hn−p−1,p+1

It gives a homomorphism,

μ : H1(X,ΘX) →⊕

Hom(Hn−p,p, Hn−p−1,p+1)

Thus,there is a homomorphism

μ0 : H1(X,ΘX)c1(L) →⊕

Hom(Pn−p,p, Pn−p−1,p+1)

Theorem 3.13 (Griffiths).(dλ)0 = μ0 ◦ ρ

where ρ = κ(0). κ is the Kodaira-Spencer map for the family f over Sc1(L).Inthe case ρ is injective ,then the infinitesimal Torelli theorem will follows from theinjectivity of μ0.

When X is algebraic manifold with trivial canonical line bundle, the first pieceof μ

H1(X,ΘX) → Hom(H0(X,ΩnX), H1(X,Ωn

X))

is naturally injective.

Corollary 3.14. If (X,L) is algebraic manifold with trivial canonical line bun-dle,then the infinitesimal Torelli theorem holds for the Kuranishi family . In partic-ular,it holds for X Calabi-Yau manifold.

Definition 3.15 (Local Torelli). we say Local Torelli Theorem holds for (X,L) ifthe differential dφ is an injection from tangent space T[X] into Tφ([X]) while [X] ∈ M

a closed point corresponds to (X,L).(For example, Calabi-Yau manifolds,etc)

Remarks 3.16. Infinitesimal Torelli theorem ⇒ Local Torelli theorem. But theconverse is not true.For example,the family of hyperelliptic curves.

Shafarevich Conjecture for Calabi-Yau Varieties 19

3.2.3. The Weil-Petersson metric on the moduli space. Let

f : X → M

be a maximal subfamily of the Kuranishi family of Calabi-Yau manifolds with afixed polarization [ω].

By the BTT theorem,the Kuranishi space of Xt is unobstructed and the Kodaira-Spencer map

ρt : TM,t → H1(Xt, TXt)ω ∼= H0,1

∂(TXt)ω

is injective everywhere along M.According to Yau’s solution to Calabi’s Conjec-ture,there is an unique Kahler Einstein( Ricci flat) metric g(t) on Xt in the givenpolarization [ω(t)]. Then g(t) induces a metric on Λ0,1(TXt),so one can define Weil-Petersson metric GWP on M :

for any v, w ∈ TM,t

GWP (v, w) :=∫

Xt

< ρt(v), ρt(w) >g(t)

Let Ω(t) ∈ Γ(Xt,∧nΩ1Xt

)) = Γ(Xt,OXt) be a flat holomorphic n-form on Xt withrespect to the K-E metric g(t), it had been shown by Tian and Todorov (cf.[29] [30])

GWP (v, w) =Q(C(i(v)Ω(t)), i(w)Ω(t))

Q(C(Ω(t)),Ω(t))= −

∫Xti(v)Ω(t) ∧ i(w)Ω(t)∫

XtΩ(t) ∧ Ω(t)

where the morphism

H1(Xt, TXt)ω → Hom(Pn,0, Pn−1,1) ∼= Pn−1,1(Xt)

via v �→ i(v)Ω(t) is an isomorphism.Here Q is the flat inner product on the VHS Pnf∗(Q).In fact,because TM is

mapped to Pn−1,1 isomorphically,the Weil-Petersson metric GWP is induced formthe Hodge metric h (i.e. h(·, ·) := Q(C·, ·)) on the n-th piece of the polarized VHSassociated to Pnf∗(Q),hence GWP is Riemannian metric on M .

Furthermore, the Kahler form of the metric GWP is

(WP) ωWP (t) = −√−12

∂∂ log h =√−12

Rich(Hn,0(Xt))

In particular,ωWP is independent of the polarization due to Hn,0 = Pn,0.

3.3. Appendix: Fundamental results of Calabi-Yau manifolds. The follow-ing is the well-known theorem of Yau Shing-Tung :

Theorem 3.17 (Calabi-Yau Theorem[38]). For any compact Kahler Manifold X ofcomplex dimension n with KX = ∧nΩ1

X = OX (i.e. c1(X) = 0), then there is anunique Ricci flat metric on M(ie. ∃ | Kahler metric gij such that R(g)ij = 0).

Definition 3.18 (Equivalence Definition of Calabi-Yau Manifold). Let X be acompact manifold.

(a) Originally, X admits a Riemannian metric with global holonomy group

0 = H∗ ⊆ SU(n).

(b) X is a compact Kahler manifold with trivial canonical bundle.

20 Yi Zhang

Now, we give a sketch description the equivalence of the definitions.

As well known,SU(n) is the subgroup of U(n) preserving an alternate complex n-form on Cn.Thus, a compact manifold X with holonomy H∗ = 0 contained in SU(n)is really a Kahler manifold (of complex dimension m) with a non-zero parallel formωX of type (n, 0)(under the Levi-Civita connection D).By standard Riemanniangeometry,one has

d =∑

�i ∧DXi

where {Xi} is moving frame of tangent bundle and {�i} is the dual frame in cotan-gent bundle,So 0 = d(ω) = ∂(ω) + ∂(ω).Thus the non-zero (n, 0)- parallel form ω is∂-closed,i.e. holomorphic.

Let ΘX be holomorphic tangent sheaf of X and Ω1X = (ΘX)∗, the canonical

bundle KX := ∧nΩ1X is flat holomorphic line bundle and ωX is the global nonzero

holomorphic section of KX .Therefore,the canonical line bundle KX is trivial. Inother words,by Calabi-Yau theorem there is unique Kahler metric on M such thatthe Ricci curvature (which for a Kahler manifold is just the curvature of KX) iszero.Thus the holonomy group H∗ is in SU(n).

Therefore, by Calabi-Yau theorem,the compact manifold X admits a metric withholonomy contained in SU(n) if and only if X is Calabi-Yau manifold. As shownabove,let X be a Calabi-Yau manifold of n dimension, then

ΘX∼= Ωn−1

X .

By the short exact sequence

0 → Z → OX → O∗X → 0,

one will obtain

· · · → H1(X,Z) → H1(X,OX) → Pic(X) c1−−→ H2(X,Z) → · · · .Because X is Calabi-Yau, so for 0 < i < n(using the following proposition 3.19),

hi(X,OX) = h0,i(X) = hi,0(X) = h0(X,ΩiX) = hn,n−i(X) = 0

Thus0 → Pic(X) → H2(X,Z)

Especially, by Lefschetz-(1, 1) Theorem

Pic(X) = H1,1Z =: H1,1

⋂H2(X,Z)

Proposition 3.19. Let X be a compact Kahler manifold with dimension n .

(I) If X has dimension n ≥ 3, with holonomy group H = SU(n),then

H0(X,ΩpX) = 0

for 0 < p < n.Moreover, X is a projective variety.(II) Let Kahler class be the set of the class in H2(X,Z) which forms Kahler

metrics,denote it K.It is obvious the Kahler cone K ⊗ R is open in

H1,1R := H1,1

⋂H2(X,R).

Thus, any compact Kahler manifold X with H0(X,Ω2X) = 0 is projective.

Shafarevich Conjecture for Calabi-Yau Varieties 21

Proof of 3.19. Sketch :(I) Because the global holomony group H∗ = SU(n), KX = ∧nΩ1

X OX .Thusone has a Ricci-flat Kalher metric by 3.17.

Let τ be any holomorphic tensor filed(covariant or contravariant), by Bochnerformula

Δ(‖τ‖2) = ‖∇τ‖2 +Ric(τ∗, τ∗) = ‖∇τ‖2

So τ is parallel.Let x ∈ X and V = Tx(X),by Borchner formula, H0(X,Ωp

X) can be identifiedwith SU(n) invariant subspace of

∧p V ∗. Because SU(V ) acts irreducibly on∧p V ∗,

the invariant space is zero unless p = 0 or p = n.(H0(X,Ωp

X) �∧p V ∗ for 0 < p < n,because the connection is not zero)

(II) Now,H2(X,C) = H1,1 and H1,1R = H2(X,R),so Kahler cone is open in

H2(X,R).Therefore, the Kahler cone contains an positive integral class [ω] andby

Pic(X) c1−−→ H2(X,Z) → H2(X,OX) = 0,this [ω] corresponds a L in Pic(X). The well-known Kodaira embedding theoremasserts :

A line bundle must be ample when its Chern Class is in Kahler class.Thus, L is an ample line bundle,i.e. X is a projective manifold.• (Continue I) From (II),when X is a Calabi-Yau manifold ,then

H0(X,Ω2X) = 0.

So X is projective . �In this Lecture, when we consider Calabi-Yau manifold, we always mean the

projective Calabi-Yau manifold,i.e.the manifold has Holomony Group

H∗ = SU(n).

Example 3.20 (Complete intersection). Let X ⊂ Pn+k be a varieties defined byF1 = F2 = · · · = Fk = 0 and degFi = di.For generic choice of Fi X is smoothmanifold of dimension n. By adjunction formula,the canonical line bundle

KX = OX(k∑1

di − n− k − 1)

is trivial if∑di = n + k + 1.By Lefschetz’s hyperplane theorem, these manifolds

satisfyH0(Ωi

X) = 0 for 0 < i < n.

Thus one obtains Calabi-Yau manifold of complete intersection type.

Example 3.21. Taking a double cover of P3 branched over 8 planes in generalposition,blowing up along the 28 singular lines,then one will obtain a Calabi-Yauthreedfold.These works are started by S.T.Yau, and in a recent processing works ofLian,Todorov and Yau,they want to show the moduli space of CY threefolds of thistype is

(G \ SU(3, 3))/S(U(3) × U(3))(It really bellowed to a conjecture of Dolgachev.I) a Shimura variety which can beembedded into the moduli stack of Abelian varieties.

22 Yi Zhang

4. Rigidity on Families of Calabi-Yau manifolds

4.1. Endomorphism of Higgs bundles over product varieties. Let V be anarbitrary polarized R-VHS over S0 × T 0 such that local monodromies around thedivisor at infinity are quasi unipotent.(it shown by Landman,Katz and Borel thatit is true for polarized Z-VHS). Extending the associated Higgs bundle E to theinfinity and one gets the quasi canonical extension Higgs bundle :

θ : E → E ⊗ Ω1S×T (logD)

One can obtain a meaningful endomorphism σ on V|St (cf. Jost-Yau Theorem in[10] and another proof of Zuo in [44]):Hint: Consider the two projections pS : S × T → S and pT : S × T → T , One has then

Ω1S×T (logD) = p∗SΩ1

S(logDS) ⊕ p∗T Ω1T (logDT ),

and the Higgs map

(4.0.1) θ : p∗SΘS(− logDS) ⊕ p∗T ΘT (− logDT ) → End(E),

is really the sheaf map of the differential of the extended period map.The restriction of the Higgs map to St, is

(4.0.2) θ|St: (p∗SΘS(− logDS) ⊕ p∗T ΘT (− logDT ))|St

→ End(E)|St.

Note that

p∗T ΘT (− logDT )|St

l⊕OSt

,

where l is dimension of T ,let 1T ∈ p∗T ΘT (− logDT )|Stbe any constant section, then one

obtains an endomorphism

(4.0.3) σ := θ|St(1T ) : E|S0

t→ E|S0

t.

As σ is coming form Higgs filed, it must be of (−1, 1) type and over C.Moreover σ is amorphism of Higgs sheaf,i.e the diagram

E|S0t

θS0

t−−−−→ E|S0t⊗ Ω1

S0t⏐⏐�σ

⏐⏐�σ⊗id

E|S0t

θS0

t−−−−→ E|S0t⊗ Ω1

S0t

is commutative because of θS0t∧ θS0

t= 0 where

θSt: ΘSt

(− logDSt) ↪→ (p∗SΘS(− logDS) ⊕ p∗T ΘT (− logDT ))|St

→ End(E)|St

the Higgs map of E|St.

Actually,it is shown by Zuo [44] that the image of the map 4.0.2 is contained inthe kernel of the induced Higgs map on End(E). Therefore, the image is a Higgssubsheaf with the trivial Higgs field. The Higgs poly-stability of End(E)|St impliesany section in this subsheaf is flat. One sees also this flat section is of Hodge type(−1, 1).

For a weight n R-VHS VR, we always have a non degenerate pairing VR × VR →R(−n),then we get (VR)∨ = VR(n),a VHS by shifted (−n,−n).

Shafarevich Conjecture for Calabi-Yau Varieties 23

Similarly,when Higgs bundle E carries R-structure,then the dual Higgs bundle is

(E∨ =⊕p+q=n

E∨−p,−q, θ∨)

where E∨−p,−q = (Ep,q)∨ = Eq,p, θ−p,−q∨ = −θq,p and

θ−p,−q∨ : E∨−p,−q → E∨−p−1,−q+1 ⊗ Ω1M

Thus we get the Higgs bundle

(End(E) =⊕r+s=0

End(E)r,s, θend)

with

(4.0.4) θend(u⊗ v∨) = θ(v) ⊗ v∨ + u⊗ θ∨(v∨).

Proposition 4.1. (Proposition 2.1 in [44]) Denote (End(E), θend) the system Hodgebundles corresponding to the polarized VHS on End(VR) which is induced by thepolarized VHS on VR.Then

θend(dφ(TM )(− logD∞)) = 0

Using the generalized Uhlenbeck-Yau-Donaldson-Simpson correspondence on higherdimensional quasi-projective manifolds(Jost-Zuo [11][43]),we obtain

Theorem 4.2 (Zhang [41],[42]). Let M be quasi-projective manifold with goodcompletion.Assume the Higgs bundle (E, ∂, θ) comes form tamed harmonic bundle(V,H,∇). Let e be a holomorphic section of (E, ∂),then θ(e) = 0 if and only if e isa flat section of (V,∇).

Remark: If M is compact Kahler manifold,the statement has already been shownby Simpson.

Exercise 4.3. The endomorphism σ obtained above is a flat (−1, 1)-type sectionof End(E)|S0

t.Therefore, σ can be seen as a endomorphism of local system C,i.e.

(4.3.1) σ : VC|S0t→ VC|S0

t.

Remark: Naturally, we have a question: What’s the conditions of family such thatσ is defined over R,more over Q?

4.2. Rigid criterion for families of Calabi-Yau manifolds. Here we shoulddeclareThe infinitesimal Torelli theorem holds for all manifolds we study in the

following sections.So, there is a natural condition for a smooth family f : X →M of n dimensional

projective manifolds not to be trivial:(∗∗) The differential of the period map for Pnf∗(C) is injective at some points

of M .The meaning of the condition:We have dim Image(η) = dimC where η is the induced modular map C → M.

We say f is rigid, if there exists no non-trivial deformation over a non-singularquasi-projective curve T 0.

24 Yi Zhang

Here a deformation of f over T 0, with 0 ∈ T 0 a base point, is a smooth projectivemorphism

g : X →M × T 0

for which there exists a commutative diagram

X �−−−−→ g−1(M × {0}) ⊂−−−−→ X

f

⏐⏐� ⏐⏐� ⏐⏐�gM

�−−−−→ M × {0} ⊂−−−−→ M × T 0

.

Observation: Let f : X → M be a smooth polarized family of n dimensionalprojective manifolds. Following the infinitesimal Torelli theorem of CY manifolds,if the deformation g of f is non-trivial,then the period map of the extended familyg is not degenerate along T 0-direction for some points of M × {0}.

At all, we obtain a criterion for rigidity from the previous subsection 4.1.

Theorem 4.4 (Criterion for Rigidity [44],[42]). Let f : X → M be a smooth po-larized family of n dimensional satisfying the condition (∗∗),if the family f is notrigid,then there is a non-zero flat section σ of End(Pnf∗(C))−1,1. And this endo-morphism comes from a flat section of σ ∈ End(E)−1,1 where (E, θ) be a Higgsbundle induced from Pnf∗(Q). Moreover, the Zariski tangent space of deformationspace of this family f is into

End(Pnf∗(C))−1,1.

One can also refer to the papers of Jost-Yau [10]and Faltings [8], to get the similarcriterion on Families of Abelian Varieties. With the same method,it is not difficultto generalize this result to nonrigid polarized VHS.

Shafarevich Conjecture for Calabi-Yau Varieties 25

4.3. Some applications of the criterion.

4.3.1. Rigidity of Lefeschetz pencils of Calabi-Yau manifolds.

Theorem 4.5 (Zhang [41] [42]). Let f : X → P1 be Lefschetz pencil such thatf : X0 → P1 \ S is smooth and smooth fibers are Calabi-Yau n-folds. Assume fsatisfies the condition (∗∗),then n must be odd.

The local system of vanishing cycles space VQ will be in Pnf∗(Q) because V isabsolute irreducible with (VC)n,0, (VC)0,n ∈ Pnf∗(C).Actually,from the Lefschetz de-composition

Rnf∗(Q) = ⊕kLkPn−2kf∗(Q)

where Pn−2kf∗(Q) = ker(Lk+1 : Rn−kf∗(Q) → Rn+k+2f∗(Q)) with the polarizationL,we have the decomposition

Pnf∗(Q) = V ⊕ (Pnf∗(Q))π1(P1\S).

Remarks 4.6. Notes on the theorem:(I) The Lefschetz pencil is just the set of hypersurfaces over a complex line,so

the singular fiber only has unique simple singularity. Lefschetz pencils havevery good properties.It should be noted that most pencils are Lefschetzpencils.One can refer to Deligne’s paper for the strict definition.

(II) The proof the theorem is dependent on the moduli properties of Calabi-Yaumanifolds and the Kazhdan-Margulis Theorem[6] which is essential inDeligne’s proof of Weil Conjecture I :

The image π1(P1 \ S) in Sp(VC, ( , )) is open.

Theorem 4.7 (Zhang [41],[42]). The above families must be rigid.

Proof. Let C0 = P1 \ S.Assume the statement is not true, we have the nontrivialextension family

X \ f−1(S) ⊂−−−−→ X

f

⏐⏐� ⏐⏐�gC0 × {0} ⊂−−−−→ C0 × T 0

.

where T 0 is a smooth quasi-projective curve.The local system Pnf∗(C) defines a polarized VHS,then we have a associated

Higgs bundle (E, θ) .

Lemma 4.8. There is a splitting of the Higgs bundle

(E, θ) = Ker(σ)⊕

(Ker(σ))⊥

for the flat (−1, 1)-type non zero endomorphism σ : E → E. The statement is alsotrue when C0 is replaced by M a higher dimensional quasi-projective variety.

proof of the lemma. First, the non-degenerate polarized Q and σ are flat and definedover C. Thus,it is not difficult to obtain the splitting of the local system over C

Pnf∗(C) = Ker(σ) ⊕ (Ker(σ))⊥

which is compatible with the polarization Q. (Ker(σ))⊥ is orthogonal componentof Ker(σ) in Pnf∗(C). It is a special case of Deligne’s complete reducibility. Then,

Pnf∗(C) ⊗OC = Ker(σ) ⊕ (Ker(σ))⊥

26 Yi Zhang

because σ is also a OC-linear map. Restricting the Hodge filtration of VHS tothese sub local system and taking the grading of the Hodge filtration,one obtains adecomposition of the Higgs bundles.Exactly,we obtain a splitting of Complex Vari-ation of Hodge Structure[27][28]. It is a example of generalized Uhlenbeck-Yau-Donaldson-Simpson correspondence theorem. �

Back to the proof of the theorem:E0,n ∈ Ker(σ) is always true along M ,but as non-triviality of the deformation ofthe family,at one point s0 of P1 \S, σ : En,0|C0 → En−1,1|C0 is injective(local Torellitheorem),thus at s0,

En,0 � Ker(σ) and E0,n ⊂ Ker(σ).

As for Lefschetz pencil,we have already shown that the fundamental representa-tion into space of the vanishing cycles

ρ : π1(P1 \ S) → Aut(VC,s0)

is irreducible and En,0 and E0,n are all in VC(4.5). It is a contradiction. �Comments 4.9. The proof of this theorem depends heavily on the structure of Lefschetzpencils,Hard Lefschetz Theorem and properties of Moduli space of Calabi-Yau manifolds. Itseems that we can generalize this statement to l− adic cohomology.We also know Lefschetzpencil is just algebraic geometry version of Morse theory. Can we find the Physics model ofthis structure,or can we find the Witten-style proof of this theorem as Laumon has reprovedthe Weil conjecture by replacing the technique of Lefschetz pencil of Deligne with the methodof deformation in Witten’s Morse theory.

Definition 4.10. Let V be a local system of Q-vector space on a quasi-projectivemanifold S with monodromy representation

ρ : π1(S, s0) → GL(V ), V := Vs0 .

1. The monodromy group π1(S, s0)mon is denoted by the Zariski closure of thesmallest algebraic subgroup of GL(V ) containing the monodromy representationρ(π1(S, s0)).

2. If V carries a nondegenerate bilinear formQ which is symmetric or anti-symmetricand which is preserved by the monodromy group. We call the monodromy is bigif the connected component of origin π1(S, s0)mon0 acts irreducibly on VC.

Theorem 4.11. Let π : X → Y a non-isotrivial smooth family of CY manifold ofdimension n over the quasi-projective manifold Y and (Rnπ∗Q)prim carries a naturepolarized VHS. Suppose that there is a sub Q-VHS V with big monodromy and thefirst Hodge piece is in VC.Then the family π : X → Y must be rigid.

4.3.2. General result on rigidity of families with strong degenerate. We assume allfamilies in this subsection satisfy the (∗∗) condition in section 4.2.

Definition 4.12. Let f : X → C be a family over a smooth projective curve C,and {c0, · · · , ck} are singular values of f .Assume f satisfies the following condition:

a) Xi = f−1(ci) = Xi1 + · · ·+Xiri is a reduced fiber of a union of transversallycrossing smooth divisors for all i i.e. f is a semi-stable family;

b) the cohomology of every smooth component of X0 has type (p, p).We say this family has strong degenerate at the singular fiber X0 = f−1(c0)

Shafarevich Conjecture for Calabi-Yau Varieties 27

Remark: The conception of strong degenerate comes from the literature of HodgeConjecture on Abelian variety.

Using the Kunneth formula,we will get the following result immediately.

Corollary 4.13. If the family f : X → C has strong degenerate property at c0,thenthe product family

π : Y � X ×C X → C

is also strong degenerate at c0.

Let M be a quasi-projective variety,(E, θ) be the Higgs bundle with positiveHermitian metric H over M .We always assume (E, θ) has R-structure.

Exercise 4.14. By Kunneth’s formula,as a vector space

(4.14.1) End(E)−1,1 ⊂ Rn+1π∗(Ωn−1Y/M )

Theorem 4.15 (Zhang [41],[42]). Strong degenerate families(not only for familiesof Calabi-Yau manifolds) over any algebraic curve must be rigid.

Proof. f : X → C is a strong degenerate family with

{c0, · · · , ck} = C − C0

the singular values of f and the fiber is a projective variety of dimension n. Let

f : X = f−1(C0) → C0

be the restricted smooth family.Define the product family

π : Y � X ×C X → C,

as shown in the above corollary,it is strong degenerate.Let Y := π−11 (C0) = X ×C0

X ,then we have the restricted smooth family

π : Y → C0

If f : X → C0 is not rigid, We will get a nonzero flat (-1,1) morphism

σ : Pnf∗(C) → Pnf∗(C)

Moreover, shown in the above lemma

σ ∈ (Rn+1π∗(Ωn−1Y/C0

))π1(C0).

Following Deligne’s Hodge theory [5],we know the commutative diagram

Hn(Y ,C) i∗−−−−−−−−−−−−−−→ Hn(Y,C)

���

i∗t

���

i∗t

Hn(Yt,C)π1(C0,t)

where it : Yt ↪→ Y, it : Yt ↪→ Y are natural embedding;and

Hn(Yt,C)π1(C0,t) ∼= H0(C0, Rnπ∗(C))

is an isomorphism of Hodge Structure. As i∗t is surjective Hodge morphism for everyt ∈ C0, we have the following restriction maps induced by Yt ⊂ Y,

(4.15.1) rp,qt : Hq(Y,Ωp

Y) ↪→ Hn(Y,C)i∗t−−→ Hn(Yt,C) → Hq(Yt,Ωp

Yt)

28 Yi Zhang

where p+ q = n.From the restriction maps, we have

Claim 4.16. The component of type (p, q) of the group Hn(Yt,C)π1(C0,t) is theimage of Hq(Y,Ωp

Y) under rp,qt .

For each nonsingular dimensional n reduced algebraic cycle B ⊂ Y and eachα ∈ Hq(Y,Ωp

Y),we consider the integral∫Bα ∧ α =

∫Bα|B ∧ α|B

then ∫Bα ∧ α = 0 ⇔ α|B = 0

WLOG, f is strong degenerate at c0, so is π. Denote

Y0 = D1 + · · · +Dk

Fixing p, q,if Hp,q(Dj) = 0 for each smooth component Dj of Y0,then

α|Dj = 0,∀jSo ∫

Y0

α ∧ α = 0.

The fiber Yt is homological equivalent to the singular Y0 for any t ∈ C0, and theform α ∧ α is closed,therefore

0 =∫Y0

(α ∧ α)|Y0 =∫Yt

(α ∧ α)|Yt

We obtain α|Yt = 0,i.e. the restriction map

rp,qt : Hq(Y,Ωp

Y) → Hq(Yt,ΩpYt

)

is zero map.Now, π is a family strong degenerate at c0, the cohomology of every smooth

component of Y0 has type only (p, p).Especially,

Hn−1,n+1(Dj) = 0

for all j.Therefore,the restriction map

rn−1,n+1t : Hn+1(Y,Ωn−1

Y ) → Hn+1(Yt,Ωn−1Yt

)

is zero map,i.e(Rn+1π∗(Ωn−1

Y/C0))π1(C0) = 0.

It is a contradiction.So the family f is rigid. �

In particular, form the theorem we get immediately

Corollary 4.17 (Weak Arakelov’s Theorem). Let f : X → C be a semi-stablefamily over a smooth projective curve C, X0 is the singular fiber

X0 = f−1(c0) = D1 + · · · +Dr.

If every Di is dominated by Pn,then this family is rigid.

Shafarevich Conjecture for Calabi-Yau Varieties 29

Remarks 4.18. If this family is a fibration of curves,then we obtain a similar resultof rigid part of Shafarevich conjecture over function field.Though it is a weak result,afamily admitting strong degenerate are in general case.

Example 4.19. All one parameter family in Pn of type

F (X0, · · · , Xn) + td∏i=1

Xτ(i) = 0

are rigid, where F is smooth homogenous polynomial with degree d.

4.3.3. Rigidity and Yukawa coupling. In this subsection, we show the relation be-tween the rigidity of Calabi-Yau manifolds and Yukawa coupling, then we give ap-plications.

• n-iterated endomorphism of σ and Yukama coupling

It is better to understand more and more about the endomorphism σ which hasdeep background in string theory.

Anyway,there is n-iterated operator

σn = σ ◦ · · · ◦ σ︸ ︷︷ ︸n

on E, and we know σk ≡ 0 if k >> 0.Furthermore, the following is held for Calabi-Yau manifolds.

Proposition 4.20. Let f : X → M be a smooth family of polarized n dimensionalCalabi-Yau manifolds satisfying the condition (∗∗).If f is not a rigid family,then then-iterated endomorphism σ must be zero.

Proof. Assume the statement is not true,we have a non zero flat endomorphismσn which is in fact a global holomorphic section of (L∗)⊗2 (under the holomorphicstructure ∂E of the associated Higgs bundle)where

L := R0f∗ΩnX/M .

Therefore (L∗)⊗2 is trivial. But it is impossible for the following reason:Let π : X → M be a maximal subfamily of the Kuranishi family of Calabi-Yau

manifolds with a fixed polarization [ω],by the BTT theorem,the Kuranishi space ofXt is unobstructed and the Kodaira-Spencer map

ρ(t) : TM,t → H1(Xt, TXt)

is injective everywhere along M.Thus, we have the commutative diagram,

X Ψ−−−−→ X⏐⏐�f ⏐⏐�πM

ψ−−−−→ M

By base change formula,

ψ∗R0π∗(ΩnX/M) = R0f∗(Ψ∗Ωn

X/M) = R0f∗ΩnX/M

30 Yi Zhang

DenoteL := R0f∗Ωn

X/M .

We have∫M

(c1(L))dimM =∫M

(c1(ψ∗R0π∗(ΩnX/M)))dimM =

∫M

(ψ∗c1(R0π∗(ΩnX/M)))dimM

Due to the works of Todorev and Tian, one has the Weil-Petersson metric on M

and the formula (cf.[29],[30])

ωWP (t) = −√−12

∂∂ logH =√−12

RicH(Hn,0(Xt))

Here H is the Hodge metric on the polarized VHS Pnf∗(C). Thus∫M

(c1(L))dimM =∫ψ(M)

(c1(R0π∗(ΩnX/M)))dimM =

∫ψ(M)

V OL(ωWP ) > 0,

Because ψ does not degenerate.�

Definition 4.21. Let f : X0 →M be smooth family of n-dimensional CY manifoldsover a quasi-projective manifold M , (E, θ) is the Higgs bundle of Pnf∗(C).

Yukawa coupling is just the n-iterated Higgs field

θn : E → E ⊗ SnΩ1M

which has deep background in string theory.Maybe you will find the definition isdifferent from other literature,but they essentially are compatible.

As the Higgs bundle (E, θ) can be splitting into

(E, θ) = (⊕p+q=n

Ep,q,⊕

θp,q)

withθp,q : Ep,q → Ep−1,q+1 ⊗ Ω1

M

Because of θ ∧ θ = 0,

θn,0 ◦ · · · ◦ θ0,n : En,0 −−→ E0,n ⊗n⊗

Ω1M

can factor throughθn : En,0 −−→ E0,n ⊗ SnΩ1

M .

So we have this definition.WLOG,the Yukama coupling can be written as

θn : SnΘM → Hom(En,0, E0,n) = ((R0f∗ΩnX0/M

)∗)⊗2

It is very interesting to understand to relations among these morphismsLet M be the quasi-projective smooth curve C0 = C \ S with S = {s1, · · · , sk}.

Using local Torelli theorem and above proposition we obtain :

Exercise 4.22 (Viehweg-Zuo & L-T-Y-Z). Let f : X0 → C0 be a smooth familyof n-dimensional Calabi-Yau manifolds satisfying the condition (∗∗). If the Yukawacoupling is not zero at some point,then the family f must be rigid.

Shafarevich Conjecture for Calabi-Yau Varieties 31

Hint1.Using the properties of Higgs bundle and locally Torelli theorem for Calabi-Yau.

2.Certainly,one can prove this criterion directly by the same method in proof ofproposition 4.20.

3.One can obtain the result right away by Viehweg-Zuo’s criterion 4.24

Example 4.23 (Liu-Todorov-Yau-Zuo). Let F ,H be homogenous polynomial ofdegree n + 2 such that F defines a nonsingular hypersurface in Pn+1 and H is notin the Jacobian ideal of F .Let us think about a special family F (t) = F + tH fort ∈ P1.If the family satisfied the condition that

Hn is not in the Jacobian ideal of F + μH for some μ ∈ P1,

as shown in the paper [18],the Yukawa coupling at the point μ ∈ P1 is not zero.Thusthis family will be rigid.

We should give a picture to this description,though Higgs field is essential theKodaira-Spencer map which determines totally the deformation of the manifold,if wedeform every manifold of the family along same direction, we just get a deformationof the family.Remark on the proposition 4.22: Let (E, ∂, θ) be the analytic Higgs bundleover a quasi projective variety M such that D∞ = M \ M normal crossing.IfHiggs map θ has regular singularities at D,E has a canonical extension over M ,byGAGA principal E becomes a algebraic vector bundle under holomorphic structure∂. Restricting to M , we get an algebraic Higgs bundle (E, θ),then θ is really aalgebraic map,so that the Yukawa coupling is algebraic θn is a global section of((R0f∗(ΩX/C0

))∗)⊗2 ⊗ SnΩC0 . Denote

Z = {s ∈ C0 | θns = 0},

then Z is either a set of finite points or C0.The criterion says that the family willbe rigid when Z is of finite points.And Z = C0 when the family is non-rigid, butthe converse is not necessary true.

A readable proof from view of differential geometric can be found in preprintof Liu-Todorev-Yau-Zuo [18].Actually,the author got idea of the proposition forma manuscript of Kang Zuo and this paper.Communication by email, Liu,K.F andTodorov again pointed out the relation between σn and Yukawa coupling from theview of solution of Picard-Fuch equation,and that one zero implies another zero too.

The proposition 4.22 is really a special case of the Viehweg-Zuo’s original works,theydeal with more general cases: not only Calabi-Yau manifolds but also any projectivemanifolds with semi-ample canonical line bundle(include CY);or of general type [33,Corollary 6.5].The proof can also be found in their another paper[35, corollary 8.4].

I would like to introduce this general criterion and show the application:

Criterion 4.24. [Viehweg-Zuo] Let Mh be the coarse moduli space of polarizedn-folds with semi-ample canonical line bundle ω; or of general type.Assume Mh

has a nice compactification and carries a universal family π : X → M(In the real

32 Yi Zhang

situation,one needs to work on stacks).Let kM be the largest integer such that kM-times iterate Kodaira-Spencer class of this deformation complex on the moduli stackM is not zero.Certainly, 1 ≤ kM ≤ dimension of the fibres.

If kM-times iterated K-S class for a family f : X → Y is not zero,thenthe family f must be rigid.

Remark: Here we always assume the flat family f : X → Y satisfies the “good”completion : X ,Y are projective manifolds,S = Y −U is a reduced normal crossingdivisor such that the restricted family f : V → U is smooth morphism and Δ =f∗S = X \U is a also a normal crossing divisor(if Δ is reduced,we get a semi-stablefamily).

We have Deligne quasi-canonical extension of the VHS Rnf∗QV (i.e. that the realpart of the eigenvalues of the residues around the components of S lies in [0, 1)),the reader can see a simple example of canonical extension soon),take grading of thefiltration,so get a extension Higgs bundle

(⊕p+q=n

Ep,q,⊕

θp,q)

whereθp,q : Ep,q → Ep−1,p+1 ⊗ Ω1

Y (logS).

Generally, we still call the n-iterated Higgs map

θn : En,0 −−→ E0,n ⊗ SnΩ1Y (logS)

be Yukawa coupling.By abusing the notation,sometimes we regard the coupling as

θn : Sn(TY (− logS)) −−→ E0,n ⊗ (En,0)∨.

When the local monodromies are all unipotent around the components of S(forexample,if f is semi-stable the condition will hold),it happens that (cf.[9])

Ep,q = Rqf∗ΩpX/Y (log Δ).

One finds for Δ = f∗S

(4.24.1) ΩnX/Y (log Δ) = ωX/Y (Δred − Δ) = ωX/Y .

The Yukawa coupling is

θn : R0f∗ΩnX/Y (log Δ) −−→ Rnf∗OX ⊗ SnΩ1

Y (logS),

so it can be regarded as

θn : SnT 1Y (− logS) −−→ Rnf∗OX ⊗ (f∗ωX/Y )−1.

The Criterion 4.24 implies the following :

Corollary 4.25. f : X → Y is a family as above and every smooth fiber is of n-dimprojective manifolds with semi-ample canonical line bundle or of general type.

If the Yukawa coupling

En,oθn

−−→ E0,n ⊗ SnΩ1Y (logS)

is not zero,then the family f : X → Y is rigid.

Shafarevich Conjecture for Calabi-Yau Varieties 33

Proof of the corollary 4.25: We give a proof in situation of the monodromies areunipotent(otherwise one can use the Kawamata covering trick to reduce the problemto the case [33]).

(4.25.1) 0 −−→ f∗Ω1Y (logS) −−→ Ω1

X (log Δ) −−→ Ω1X/Y (log Δ) −−→ 0

The wedge product sequences

(4.25.2) 0 −−→ f∗Ω1Y (logS) ⊗ Ωp−1

X/Y (log Δ) −−→gr(Ωp

X (log Δ)) −−→ ΩpX/Y (log Δ) −−→ 0,

where

gr(ΩpX (log Δ)) = Ωp

X (log Δ)/f∗Ω2Y (logS) ⊗ Ωp−2

X/Y (log Δ).

For the invertible sheaf L = ΩnX/Y (log Δ) = ωX/Y we consider the sheaves

F p,q := Rqf∗(ΩpX/Y (log Δ) ⊗ L−1)

together with the edge morphisms

τp,q : F p,q −−→ F p−1,q+1 ⊗ Ω1Y (logS),

induced by the exact sequence (4.25.2), tensored with L−1. As explained in [33],Proof of 4.4 iii), F p,q = Rqf∗(

∧n−p TX/Y (− log Δ)) can be regarded as the deforma-tion complex of f : X → Y .Moreever over U the edge morphisms τp,q can also beobtained in the following way. Consider the exact sequence

0 −−→ TV/U −−→ TV −−→ f∗TU −−→ 0,

and the induced wedge product sequences

0 −−→n−p+1∧

TV/U −−→ Tn−p+1V −−→

n−p∧TV/U ⊗ f∗TU −−→ 0,

where Tn−p+1V is a subsheaf of

∧n−p+1 TV . One finds edge morphisms

τ∨p,q : (Rqf∗(n−p∧

TV/U )) ⊗ TU −−→ Rq+1f∗(n−p+1∧

TV/U ).

Restricted to η ∈ U those are just the wedge product with the Kodaira-Spencerclass. Moreover, tensoring with Ω1

U one gets back τp,q|U .Because of Rqf∗(

∧n−p TX/Y (− log Δ)) = Rqf∗(ΩpX/Y (log Δ)⊗ω−1

X/Y ),we also call

τn−q,q : F p,q −−→ F p−1,q+1 ⊗ Ω1Y (logS)

the log Kodaira-Spencer map and

τm : Fn,0 = OYτn,0−−→ Fn−1,1 ⊗ Ω1

Y (logS)τn−1,1−−−−→ Fn−2,2 ⊗ S2(Ω1

Y (logS)) −−→ · · ·τn−m+1,m−1−−−−−−−−→ Fn−m,m ⊗ Sm(Ω1

Y (logS))

m-times iterated K-S class.One has a factor map

34 Yi Zhang

Sn(TY (− logS))

���τn(f) �

��θn(f)

Rnf∗(ω−1X/Y ) −−−−−−−−−−−−−−→ Rnf∗OX ⊗ (f∗ωX/Y )−1

Henceθn(f) = 0 =⇒ τn(f) = 0 =⇒ τnM = 0 =⇒ kM = n.

By the Viehweg-Zuo criterion 4.24, the family f is rigid.�

Remark : Except rankEn,0 = 1,it seems difficult to get a proof only using Variationof Hodge Structure.

• Local study of families admitting maximal unipotent degenerate

Let f : X → C be family smooth over C0, (V,∇) be integrable algebraic vectorcome from the polarized VHS of smooth part of the family. (V,∇) admits quasi-unipotent monodromy,thus we have Deligne’s quasi-unipotent extension (V,∇),

∇ : V → V ⊗ Ω1C(logS),

This algebraic vector bundle corresponds to a regular filtered Higgs bundle (E, θ)α.Locally, It is explained as following:Restricting the family f : X → C to the unit disk Δ,We think about the local

degenerated familyf : Xloc → (Δ, t)

Now,WLOG we assumef ′ = f |X ∗

loc: X ∗

loc → (Δ∗, t)

is smooth and the local monodromy is unipotent,i.e.

(4.25.3) (T − 1)k+1 = 0 and (T − 1)k = 0 for a fixed integer k, 0 ≤ k ≤ n.

If k = n, T is the maximal unipotent monodromy. Let

N = log T =k∑j=1

(1/j)(−1)j(T − Id)j ,

we have Nk = 0 and Nk+1 = 0.In this simple case,let the Φ : Δ∗ → D/Γ be the period map corresponding to the

local VHS VC = Pnf ′∗(C). Before Schmid’s Nilpotent orbit theorem,Deligne alsoshowed that the holomorphic map

Ψ(t) = exp(− log t2π

√−1

N)Φ(t)

can extend cross zero.Therefore we have the Deligne canonical extension (V,∇)where V is generated holomorphically by

v(t) = exp(− log t2π

√−1

N) · v

Shafarevich Conjecture for Calabi-Yau Varieties 35

for all v ∈ Pnf ′∗(C) and v(te2π√−1) = v(t). However, as we have shown in the

previous chapter, V|0 can be represented by VU for a small neighborhood( in complextopology) U of 0.[27] Thus the monodromy T can be extended naturally to anendomorphism on E and we have the restriction and we have

T0 : V|0 −−→ V|0T0 has same property 4.25.3 as T .

Definition 4.26 (Residues of (V,∇)). For an integrable logarithmic connection

∇ : V −−→ V ⊗ Ω1Δ(log 0)

we have the composed map

(idV ⊗R0) ◦ ∇ : V ∇−−→ V ⊗ Ω1Δ(log 0)

idV⊗R0−−−−−→ V|0 ,where R0 : Ω1

Δ(log 0) −−→ C is defined by R0(fdt/t) = f(0).This composed map is zero at tV,then it induces a residue map along 0

Res0(∇) : V|0 −−→ V|0 .

Remark 4.27. Generally,letM \M = D be a reduced smooth connected divisor.Forany integrable logarithmic connection

∇ : F −−→ F ⊗ Ω1M

(log D)

we will get

(idF ⊗R0) ◦ ∇ : F ∇−−→ F ⊗ Ω1M

(log D) idF⊗R0−−−−−→ OD ⊗F ,

(idF ⊗R0) ◦ ∇ is OM−linear morphism and factors through

F restr.−−−→ F ⊗ODResD(∇)−−−−−→ F ⊗OD.

Thus ResD(∇) ∈ End(F) ⊗OD and we have the short sequence

0 −−→ Ω1M

⊗ End(F) −−→ Ω1M

(logD) ⊗ End(F) RD⊗id−−−−→ OD ⊗ End(F) −−→ 0.

Therefore by directly calculation, we have the

Exercise 4.28. If (V,∇) is induces from the VHS of the above local family,then

(4.28.1) Res0(∇) =−1

2π√−1

N.

which is a particular case of Theorem II,3.11 in Deligne’s book [4].

Definition 4.29 (Residues of Higgs map [27]). Let (V,H,D) is tamed harmonicbundle, it induces (E, θ)α a regular filtered Higgs system which includes (E, θ) aHiggs bundle over Δ∗ and a system of decreasing filtered sheaves Eα,0 which extendE across 0 and extend the Higgs map

θ : Eα,0 −−→ Eα,0 ⊗ Ω1Δ(log 0)

Here the coherent sheaf Eα is generated by e ∈ E|Δ∗ such that |e(t)|H ≤ C|t|α+ε

for every ε > 0.Denote E = E0 = ∪Eα,0

(idE ⊗R0) ◦ θ : E θ−−→ E ⊗ Ω1Δ(log 0)

idE⊗R0−−−−−→ E|0 ,

36 Yi Zhang

this morphism inducesRes0(θ) : E|0 −−→ E|0.

Also, formally we have the short exact sequence:

0 −−→ Ω1Δ ⊗ End(E) −−→ Ω1

Δ(log 0) ⊗ End(E) R0⊗id−−−−→ End(E)|0 −−→ 0.

Thus Res0(θ) = (R0 ⊗ id)(θ).

Remark: In this case (E, θ) is weight n Hodge bundle, because of θn+1 ≡ 0, wehave (Res0(θ))n+1 = 0.Then Res0(θ) is automatically nilpotent.

Proposition 4.30 (Simpson[27],Schmid[25]). The nilpotent parts of Res0(∇) is iso-morphic to the nilpotent part of Res0(θ) under E|0 ∼= V|0.In the case of Hodge bundleshown by Schmid,with the above lemma

Res0(θ) ∼= Res0(∇) ∼= −12π

√−1

N,

where N is the nilpotent part of the monodromy.

Now, we come back to the Yukawa coupling.

Exercise 4.31. If the local family f : Xloc → (Δ, t) degenerate at 0 with maximalunipotent monodromy,then the Yukawa coupling is nonzero.

Therefore,as an application of the proposition 4.22,we have an interesting resultwhich is implied in the paper of Liu-Todorov-Yau-Zuo(cf [18]) and the manuscriptof Zuo Kang[45].

Theorem 4.32 (Zhang [41][42]). Let f : X → C be a family of n-dimensionalCalabi-Yau manifolds satisfying the condition (∗∗) in the section 4.2.

If the family f admits a degeneration with maximal unipotent monodromy,then fmust be rigid.

Remark 4.33. We should point out the degenerations of Lefschetz pencil of CYmanifolds are all of minimal unipotent monodromy. So, we will ask the question:Whether the CY family with minimal unipotent monodromy degeneration is rigid?

Example 4.34. In the recent paper of Lian-Todorov-Yau,the authors show thefollowing type CY families have the degenerate of maximal unipotent monodromy(cf.[17]),thus they will be rigid by the theorem 4.32.

One parameter family of complete intersections of CY manifolds in Pn+k for n ≥ 4and k ≥ 1 defined by the following equations :

G1,t = tF1 −∏n1i=0 xi = 0, .., Gk,t = tFk −

∏nkj=n1+..+nk−1

xj = 0, t ∈ P1

where the system F1 = .. = Fk = 0 defines a non singular CY manifolds,ni = degFi ≥ 2 and

∑ni = n+ k + 1

and x i are the standard homogeneous coordinates in Pn+k.

The condition∑ni = n+ k + 1 implies that the fibers π−1(t) = Xt for t = 0 are

CY manifolds of complex dimension n.

Example 4.35. As a special case of Lian-Todorov-Yau, Morrison.D showed theYukawa coupling of the following family is not zero at 0.

X50 +X5

1 +X52 +X5

3 +X54 − 5λX0X1X2X3X4 = 0

Shafarevich Conjecture for Calabi-Yau Varieties 37

for λ ∈ P1 and [X0, X1, X2, X3, X4] ∈ P4.By the example 4.34,this family admits a degeneration with maximal unipotent

monodromy at 0.One also can show this family rigid by the example 4.23,because(X0X1X2X3X4)3 is not in the Jacobian ideal of

X50 +X5

1 +X52 +X5

3 +X54 .

Example 4.36. We should point out these two examples are both of strong degen-eration(with surgery of semi-stable reduction).Thus they have to be rigid by theorem4.15.Recently Viehweg-Zuo construct some meaningful families with maximal unipo-tent degeneration by covering trick (not only just by complete intersection)[36].

38 Yi Zhang

References

[1] Arakelov, S. Ju.: Families of algebraic curves with fixed degeneracies. (Russian) Izv. Akad.Nauk SSSR Ser. Mat. 35 (1971), 1269–1293

[2] Parsin, A. N.: Algebraic curves over function fields. I. (Russian) Izv. Akad. Nauk SSSR Ser.Mat. 32 1968 1191–1219

[3] Bedulev,E. and Viehweg,E.: On Shafarevich Conjecture for Surfaces of General Type overFunction Fields, Inv. Math. 139(2000), 603-615, Math.AG/9904124.

[4] Deligne, P. : Equations differentielles a points singuliers reguliers. (French) Lecture Notesin Mathematics, Vol. 163. Springer-Verlag, Berlin-New York, 1970.

[5] Deligne, P.: Theorie de Hodge II. I.H.E.S. Publ. Math. 40 (1971) 5–57[6] Deligne, P.: La conjecture de Weil I. I.H.E.S 43 (1974),273-307[7] Eyssidieux,P.; Mok, Ngaiming : Characterization of certain holomorphic geodesic cycles

on Hermitian locally symmetric manifolds of the noncompact type. Modern methods incomplex analysis (Princeton, NJ, 1992), 79C117, Ann. of Math. Stud., 137, Princeton Univ.Press, Princeton, NJ, 1995.

[8] Faltings,G.: Arakelov’s Theorem for Abelian Varieties, Inv. Math. 73(1983) 337-347[9] Griffths, P.: Topices in transcendental algebraic geometry. Ann of Math. Stud. 106 (1984)

Princeton Univ. Press. Princeton, N.J.[10] Jost,J; Yau, S.T.: Harmonic mappings and algebraic varieties over function fields. Amer. J.

Math. 115 (1993), No. 6, 1197–1227[11] Jost, J; Zuo, K. : Harmonic maps of infinite energy and rigidity results for representations

of fundamental groups of quasiprojective varieties. J. Differential Geom. 47 (1997), no. 3,469–503

[12] Jost, J; Zuo, K.: Arakelov type inequalities for Hodge bundles over algebraic varieties, Part1: Hodge bundles over algebraic curves. J. Alg. Geometry. Vol. 11(2002) 535-546

[13] Kovacs,S.J.: Families Over Base with Birationally Nef Tangent Bundle, J. Reine Angew.Math. 487 (1997), 171-177

[14] Kovacs,S.J. : Logarithmic vanishing theorems and Arakelov-Parshin boundedness for sin-gular varieties. Compositio Math. 131 (2002), no. 3, 291–317

[15] Liu, Kefeng: Remarks on the geometry of moduli spaces. Proc. Amer. Math. Soc. 124(1996), no. 3, 689–695.

[16] Liu, Kefeng: Geometric height inequalities. Math. Res. Lett. 3 (1996), no. 5, 693–702.[17] Lian,B.H; Todorov,A; Yau,S.T.: Maximal Unipotent Monodromy for Complete Intersection

CY Manifolds, Math.AG/0008061[18] Liu,K.,Todorov,A.,Yau,S.T.,Zuo,K.: The Analogue of Shafarevich’s Conjecture for Some

CY Manifolds, Math.AG/0308209[19] Migliorini,Li.:A Smooth Family of Minimal Surfaces of General Type over a Curve of Genus

at Most One is trivial J. Alg. Geom. 4(1995) 353-361[20] Mok, Ngaiming: Aspects of Kahler geometry on arithmetic varieties. Several complex vari-

ables and complex geometry,Part 2 (Santa Cruz, CA, 1989), 335–396, Proc. Sympos. PureMath., 52, Part 2, Amer. Math. Soc., Providence, RI, 1991.

[21] Mumford, D.: Geometric invariant theory. (1965) Second enlarged edition: Mumford, D.and Fogarty, J.: Ergebnisse der Mathematik (3. Folge) 34 (1982) Springer, Berlin HeidelbergNew York

[22] Noguchi,J. : Moduli spaces of holomorphic mappings into hyperbolically imbedded complexspaces and locally symmetric spaces. Invent. Math. 93 (1988), no. 1, 15–34

[23] Oguiso, K., Viehweg, E.: On the isotriviality of families of elliptic surfaces. preprintMath.AG/9912100, to appear in J. Alg. Geom

[24] Peters,C.:Rigidity for Variations of Hodge Structures and Arakelov-type Finiteness Theo-rems, Compositio Math. 75(1990) 113-126

[25] Schmid, W.: Variation of Hodge structure: The singularities of the period mapping. Invent.math. 22 (1973) 211–319

[26] Shafarevich,I. R.:Collected Mathematical Papers, New York: Springer-Verlag (1989).[27] Simpson, C.: Harmonic bundles on noncompact curves. Journal of the AMS 3 (1990) 713–

770[28] Simpson, C.: Higgs bundles and local system,Publ.math.IHES. 75 (1992) 5-95

Shafarevich Conjecture for Calabi-Yau Varieties 39

[29] Tian,G. :Smoothness of the Universal Deformation Space of Calabi-Yau Manifolds andits Petersson-Weil Metric. Math. Aspects of String Theory, ed. S.-T.Yau, World Scientific(1998), 629-346

[30] Todorov,A. :The Weil-Petersson Geometry of Moduli Spaces of SU(n) ≥ 3 (Calabi-YauManifolds) I. Comm. Math. Phys. 126 (1989), 325-346

[31] Viehweg,E.: Quasi-projective Moduli for Polarized Manifolds. Ergebnisse der Mathematik,3. Folge 30 (1995), Springer Verlag, Berlin-Heidelberg-New York

[32] Viehweg,E.;Zuo,K.: On the isotriviality of families of projective manifolds over curves, J.Alg. Geom. 10(2001) 781-799;or Math.AG/0002203

[33] Viehweg,E.;Zuo,K.: Base Spaces of Non-Isotrivial Families of Smooth MinimalModels,Complex geometry (Gottingen, 2000), 279–328, Springer, Berlin, 2002 orMath.AG/0103122.

[34] Viehweg,E.;Zuo,K.: On The Brody Hyperbolicity of Moduli Spaces for Canonically Polar-ized Manifolds,Duke Math Journal Vol.118 No.1(2003),103-150. Or Math.AG/0101004.

[35] Viehweg,E.;Zuo,K.Discreteness of minimal models of Kodaira dimension zero and subvari-eties of moduli stacks Math.AG/0210310

[36] Viehweg,E.;Zuo,K.: Complex multiplication, Griffiths-Yukawa couplings, and rigidy for fam-ilies of hypersurfaces. Preprint (2003) Math.AG/0307398

[37] Wolpert,S.:Chern forms and the Riemann tensor for the moduli space of curves, Invent.Math. 85 (1985), 119–145.

[38] Yau,S.T.: Calabi’s conjecture and some new results in algebraic geometry. Proc. Nat. Acad.Sci. U.S.A. 74 (1977), no. 5, 1798–1799.

[39] Yau,S.T.: A Generalized Schwarz Lemma for Kahler Manifolds, A. J. Mathematics, vol.100, Issue 1(1978), 197-203.

[40] Zhang,Qi: Holomorphic One form on Projective Manifolds,J. Alg. Geom. 6(1997) 777-787[41] Zhang,Yi: On Families of Calabi-Yau manifolds,Ph.D Thesis,IMS CUHK 2003.[42] Zhang,Yi: The Rigidity of Families of Projective Calabi-Yau manifolds.Preprint(2003)

Math.AG/0308034[43] Zuo,K :Representations of fundamental groups of algebraic varieties. Lecture Notes in Math-

ematics, 1708. Springer-Verlag, Berlin, 1999.[44] Zuo,K. : On the Negativity of Kernels of Kodaira-Spencer Maps on Hodge Bundles and

Applications. Asian Journal of Mathematics, Vol. 4,(2000), No.1, 279-302[45] Zuo.K. : On Families of projective manifolds(joint works with Eckart Viehweg), manisctipt,

a full time talk at France-Hong Kong Geometry Conference, Feb 18-Feb 22, 2002, HongKong.